Download Day23,Oct24: Time Varying Fields

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Time varying sources: Coupling E and B fields
• Maxwell’s equations need to be modified for time-varying sources and fields. The divergence equations stay the same, but the curls change. The curls of the electric and
magnetic fields each picks up a source time proportional to the time-derivative of the
other, with an additional difference in sign between them.
• Thus, time-varying electric fields produce magnetic fields, and vice-versa. This couples
two separate fields of study, electricity and magnetism, to produce electromagnetism. The
alliance between these two fields is rooted deep in relativity theory. (You can turn a static
charge into a dynamic one simply by moving relative to it, thus turning E fields into B
• The divergence equations and the boundary conditions stay the same for time-dependent
fields. The curl changes, meaning the definition of the electrostatic potential needs to be
• The electric field can now curl as well, the curl being proportional to the negative time
derivative of the magnetic induction (Faraday’s law). This is an alternate way of stating
that the induced EMF is given by LdI/dt, an equation known to all circuit theorists.
Lenz’s law further stipulates the direction of this EMF, i.e., the induced voltage creates
a current that tries to oppose the magnetic flux in order to conserve energy (i.e., creating
a field in the same direction as a decreasing field, or opposing an increasing field, etc).
• The changing flux responsible for the EMF can be realized by changing the field through
an area, by increasing or decreasing the area, or by moving the area element. This can
also be rationalized in Hterms of the force acting on electrons, creating a field and therefore
~ · d~l.
an EMF voltage V = (~u × B)
• The induction principle explains the operation of motors, generators and transformers.